Stability Analysis in Continuous- and
Discrete-Time, using the Cayley Transform
Niels Besseling
Hans Zwart
Department of Applied Mathematics
University of Twente
Control of Distributed Parameter Systems, 2009
Introduction
Our Result
Outline
1
Introduction
Continuous-Time and Discrete-Time
Known Estimates
2
Our Result
Stability
Equivalence Classes
Main Result
Niels Besseling, Hans Zwart
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Continuous-Time and Discrete-Time
Known Estimates
Cayley Transform
Let A generate a semigroup in the space X.
The Cayley transform of A is:
Ad := (I + A) (I − A)−1 .
Application
Numerical scheme, Crank-Nicolson,
Maps continuous-time to discrete-time.
Niels Besseling, Hans Zwart
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Continuous-Time and Discrete-Time
Known Estimates
Continuous vs. Discrete
Continuous-Time
Differential equation:
ẋ(t) = Ax(t)
Solutions:
Discrete-Time
Difference equation:
xd (n + 1) = Ad xd (n)
Solutions:
x(t) = eAt x0
xd (n) = And x0
Questions:
Approximation,
Stability.
Niels Besseling, Hans Zwart
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Continuous-Time and Discrete-Time
Known Estimates
Stability
A semigroup (eAt )t≥0 is strongly stable, so
eAt x0 → 0,
as t → 0.
What can be said about And for n ∈ N?
Niels Besseling, Hans Zwart
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Continuous-Time and Discrete-Time
Known Estimates
Finite-Dimensional Spaces
Let X be a finite-dimensional space with dimension s, e.g. Rs .
If the semigroup is bounded:
keAt k ≤ M,
for t ≥ 0.
Result
Then for the Cayley transform the following estimate holds:
kAnd k ≤ min(s, n + 1) e M < ∞,
for all n ∈ N.
This holds for arbitrary norm on X.
Niels Besseling, Hans Zwart
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Continuous-Time and Discrete-Time
Known Estimates
Infinite-Dimensional Spaces
Let X be a Banach space, e.g. L∞ (0, 1).
If the semigroup is bounded:
keAt k ≤ M,
for t ≥ 0.
Result
Then for the Cayley transform the following estimate holds:
√
kAnd k ≤ mM n,
for all n ∈ N+ .
Niels Besseling, Hans Zwart
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Continuous-Time and Discrete-Time
Known Estimates
Contraction Semigroups
Let X be a Hilbert space.
If the semigroup is a contraction semigroup:
keAt k ≤ 1,
for t ≥ 0.
Result
Then for the Cayley transform the following estimate holds:
kAnd k ≤ 1,
Niels Besseling, Hans Zwart
for all n ∈ N.
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Continuous-Time and Discrete-Time
Known Estimates
Analytic Semigroups
Let X be a Hilbert space.
The semigroup is a analytic semigroup.
Result
If the semigroup is strongly stable,
then the Cayley transform is also strongly stable.
Niels Besseling, Hans Zwart
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Stability
Equivalence Classes
Main Result
Overview of Our Result
eAt
Niels Besseling, Hans Zwart
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Stability
Equivalence Classes
Main Result
Overview of Our Result
eAt
Niels Besseling, Hans Zwart
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Stability
Equivalence Classes
Main Result
Overview of Our Result
eAt
Niels Besseling, Hans Zwart
Ad
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Stability
Equivalence Classes
Main Result
Overview of Our Result
eAt
Niels Besseling, Hans Zwart
Ad
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Stability
Equivalence Classes
Main Result
Overview of Our Result
Cayley
eAt
Niels Besseling, Hans Zwart
Ad
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Stability
Equivalence Classes
Main Result
Similar Stability Properties Within Each Class
Continuous-Time
eAt
Let
and
class, then
eÃt
Discrete-Time
be in the same
If eAt is stable,
then
eÃt
is stable.
Niels Besseling, Hans Zwart
Let Ad and Ãd be in the same class,
then
If Ad is stable,
then Ãd is stable.
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Stability
Equivalence Classes
Main Result
Construction of Classes
Semigroups eAt and eÃt belong to the same equivalence class, if
Z ∞
1
k(eAt − eÃt )x0 k2 dt < ∞,
for all x0 ∈ X.
t
0
Cayley Transforms Ad and Ãd belong to the same class, if
∞
X
k(Akd − Ãkd )x0 k2
k=1
Niels Besseling, Hans Zwart
1
< ∞,
k
for all x0 ∈ X.
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Stability
Equivalence Classes
Main Result
Cayley Transform Preserves Classes
Let (eAt )t≥0 and (eÃt )t≥0 be semigroups,
let Ad and Ãd be the corresponding Cayley transforms.
Then
Lemma
Z
∞
At
Ãt
k(e − e )x0 k
0
21
t
dt =
Niels Besseling, Hans Zwart
∞
X
1
n=1
n
k(And − Ãnd )x0 k2 .
Stability analysis in continuous- and discrete-time
Stability
Equivalence Classes
Main Result
Introduction
Our Result
Sketch of Proof Lemma
Lemma
Z
∞
At
Ãt
k(e − e )x0 k
0
21
t
dt =
∞
X
1
n=1
n
k(And − Ãnd )x0 k2 .
There exist functions qn , “Laguerre polynomials”, such that
Z ∞
qn (t)eAt x0 dt = And x0 − x0 ,
x0 ∈ X.
0
If {em } is an o.n. basis in X, then the functions √qnn em form an
o.n. basis on the function space with inner product:
Z ∞
hf , giH =
hf (t), g(t)it dt.
0
Niels Besseling, Hans Zwart
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Stability
Equivalence Classes
Main Result
Main Result
Let (eAt )t≥0 and (eÃt )t≥0 be semigroups,
let Ad and Ãd be the corresponding Cayley transforms.
and:
Z
0
∞
1
k(eAt − eÃt )x0 k2 dt < ∞,
t
for all x0 ∈ X.
Then
Result
Ad is strongly stable,
⇐⇒
Ãd is strongly stable.
Niels Besseling, Hans Zwart
Stability analysis in continuous- and discrete-time
Introduction
Our Result
Stability
Equivalence Classes
Main Result
Application of Our Result
Cayley
eAt
Niels Besseling, Hans Zwart
Ad
Stability analysis in continuous- and discrete-time